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If x or y is odd, then a(xy) = a(x) + a(y).

If x and y are both even, then a(xy) = a(x) + a(y) + 1.

Equivalently, if we define a function D by D(x) = a(x) if x is odd and D(x) = a(x) + 1 if x is even, then D is completely additive.

L[n_] := If[EvenQ[n], (1/3) If[n > 1, n Times @@ (1 + 1/(Select[Divisors[n], PrimeQ])), 1], If[n > 1, n Times @@ (1 + 1/(Select[Divisors[n], PrimeQ])), 1]]; Table[Length@NestWhileList[L, n, # != 1 &] - 2, {n, 2, 260}]

EvenQ[n], (1/3) If[n > 1,

n*Times @@ (1 + 1/(Select[Divisors[n], PrimeQ])), 1],

If[n > 1, n*Times @@ (1 + 1/(Select[Divisors[n], PrimeQ])), 1]]

Table[Length@NestWhileList[L, n, # != 1 &]-2, {n, 2, 260}]

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If x or y is odd, then a(xy)=a(x)+a(y).

If x and y are both even, then a(xy)=a(x)+a(y)+1.

If x or y is odd, then a(xy)=a(x)+a(y). If x and y are both even, then a(xy)=a(x)+a(y)+1. Equivalently, if we define a function D by D(x)=a(x) if x is odd and D(x)=a(x)+1 if x is even, then D is completely additive.

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(PARI) a253629(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^(f[i, 2]-1)*if(f[i, 1]>2, f[i, 1]+1, 1)) ;

a(n) = my(nb = 0); my(m = n); while(m != 2, m = a253629(m); nb++); nb; \\ Michel Marcus, Jan 21 2015

Cf. A253629.

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0, 2, 1, 3, 2, 3, 2, 4, 3, 4, 3, 4, 3, 5, 3, 5, 4, 5, 4, 5, 4, 5, 4, 6, 4, 6, 4, 6, 5, 5, 4, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 7, 5, 6, 5, 6, 6, 7, 5, 7, 6, 7, 5, 7, 6, 7, 6, 6, 5, 7, 5, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 8, 6, 7, 6, 7, 6, 8, 6, 7, 6, 8, 6, 8, 6, 8, 7, 7, 6, 7, 6, 8, 6, 7, 6, 8, 7, 8, 7, 7, 6, 8, 7, 8, 7, 8, 7, 8, 6, 8, 7, 8, 7, 8, 7, 8, 7, 8, 6, 8, 6, 9, 7, 7, 6, 8, 7, 8, 7, 8, 7, 9, 7, 8, 7, 8, 7, 8, 7, 8, 7, 9, 7, 8, 7, 9, 8, 8, 7, 9, 7, 8, 7, 8, 7, 9, 7, 8, 8, 8, 7, 9, 7, 8, 7, 8, 8, 9, 7, 9, 8, 9, 7, 9, 8, 9, 8, 8, 7, 8, 7, 9, 7, 9, 7, 9, 8, 8, 7, 8, 7, 9, 7, 9, 8, 9, 8, 9, 8, 9, 8, 9, 7, 9, 7, 9, 8, 9, 8, 9, 8, 9, 8, 8, 8, 9, 8, 9, 8, 8, 7, 10, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 8, 10, 7, 9, 8, 9, 7, 9, 9, 9, 8, 9, 7, 10, 7, 9, 8, 9, 8

Colin Defant, <a href="http://arxiv.org/abs/1501.00971">An arithmetic function arising from the Dedekind $\psi$ function</a>, arXiv:1501.00971 [math.NT], 2015.

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